Modal Identification with Mobile Sensors Using Cohen’s Class Time-Frequency Distributions

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<span style="text-align: center; font-size: 75%;">'''Modal Identification with Mobile Sensors Using Cohen’s Class Time-Frequency Distributions'''</span></div>
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'''Abstract:''' '' Dynamic characterization of structures from field measurements is useful for different purposes (e.g. retrofit validation, model updating, structural health monitoring, etc.). The identification of high spatial density mode shapes has been recently a challenge tackled using mobile sensors. These sensors travel over the structure and continuously acquire vibration data that is used to identify modal coordinates with a higher spatial density than can generally be obtained using a limited number of stationary sensors. The recorded signal from a mobile sensor is non-stationary, thus, it has significant variations in its spectral content over time, requiring a suitable processing to extract the properties not only for the time but also for the frequency domain. In this paper, Cohen’s class Time-Frequency Distributions (TFD) are proposed for the output-only dynamics identification of structures based on non-stationary signals recorded with mobile sensors. Identification is achieved through cross-time-frequency estimators using Smoothed Pseudo-Wigner-Ville (SPWVD) distribution. Results from numerical simulations using a simply supported beam subject to ambient vibration are shown and the sensibility of the proposed identification to the presence of measurement noise is evaluated. Numerical results show that use of the cross-time-frequency estimators is effective in extracting modal properties of the structures and filtering noise.''

'''Keywords:''' ''Modal Identification, Mobile Sensors, Non-Stationary Signals, Time-Frequency Distributions''

==1. Introduction==

Structural Health Monitoring (SHM) Systems usually play a crucial duty in early failure or damage detection for construction, maintenance or catastrophe management. Civil infrastructure and many other sectors can greatly benefit from systems that monitor behaviour under normal operating conditions,  without the need of interrupting regular day-to-day activity (Brownjohn, 2007; Karbhari et al., 2009).

The future trend in instrumentation for SHM is the use of smart mobile sensing along with wireless intelligent networks, as they not only provide similar information to conventional cabled systems, but can also be installed at a much lower cost and process data autonomously using their embedded microprocessors and software (Lynch, 2006; Spencer et al., 2004). Modal shapes are traditionally identified with stationary sensors fixed at locations that provide representative structural responses, however, these schemes contain restricted spatial information (Figure 1). If more spatial and temporal data is obtained from a vibrating structure, the structure’s response can be evaluated more accurately, and hence the advantage of implementing the structure with as many stationary sensors as possible during the acquisition of data. However, despite the improved accuracy in estimations, these or dense sensor array approaches can be impractical for many reasons including the cost of sensors and setup time, memory constrains, network reliability, power requirements, and physical limitations due to structure geometries (Magalhães et al., 2008, 2009; Matarazzo T. J. and Pakzad, S. N. 2013). Mobile sensing is a novel alternative in Structural Health Monitoring (SHM). The objective is to solve failings from fixed sensor array setups. Few sensors are usually used to collect dense spatial information in mobile sensing setups. Mobile sensing provides several advantages over schemes using stationary sensors, but the main advantage is that a single mobile sensor can be used to record signals continuously along the structure. Modal coordinates are identified with a higher spatial density than using stationary sensors because the mobile sensor continuously records vibration data as it travels along trajectories on the structure (Figure 2). Mobile sensing is not only easier to implement, but also more cost effective when compared to dense stationary sensor arrays (Marulanda et al., 2016; Marulanda, J., 2014; Matarazzo T. J.  and Pakzad S. N., 2013).

<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">
<span style="text-align: center; font-size: 75%;"> [[Image:Marmolejo_et_al_2019b-image2.png|600px]] </span></div>

<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">
<span style="text-align: center; font-size: 75%;">Figure 1. Spatial resolution of mode shapes: stationary sensors</span></div>

<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">
<span style="text-align: center; font-size: 75%;"> [[Image:Marmolejo_et_al_2019b-image3.png|600px]] </span></div>

<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">
<span style="text-align: center; font-size: 75%;">Figure 2. Spatial resolution of mode shapes: mobile sensors</span></div>

Recent implementation of mobile sensor networks has been diverse, although limited, and is still under development. Zhu et al. (2010, 2012) proposed a sensing device. Such device will record data at selected nodes, but will not do so while the sensor is moving.  Sibley et al. (2002) and Dantu et al. (2005) implemented Robomote, miniature low cost robots for mobile sensors networks. Partial modal identification include the use of a moving vehicle to identify frequencies of a single span bridge using frequency domain techniques (Cerda F. et al., 2012; Lin and Yang, 2005). Marulanda et al. (2016) and Marulanda (2014) developed a novel technique that uses two sensors, one mobile and the other stationary, in which the mobile sensor records continuously, and, in a subsequent procedure, performs the modal identification through spectrograms. Matarazzo et al. (2014, 2016) proposed a method of collecting data through continuous mobile detection in the presence of missing time and space observations. As the corresponding matrices have incompatible data records, they developed and successfully implemented an algorithm, called STRIDE, to cover for such “empty spaces”.

The use of mobile sensors, typically accelerometers, require an analysis of non-stationary response signals. Time-Frequency analysis has been proposed by several authors to analyse non-stationary responses in structural identification and in the assessment of mechanical damage. Staszewski et al. (1994) and Rioul et al. (1991) have  devoted several papers to modal identification using  the Short-Time Fourier Transform (STFT) and Wavelet Transform (WT).  The STFT transforms the signal in a two-dimensional time-frequency plane, assuming that the signal is stationary when viewed through a limited extension window. The Fourier Transform of the windowed signal leads to a time-frequency distribution (Gabor, 1946; Arango, 2009; Oppenheim, 2010). The STFT is linear and therefore provides poor energy information about the signal. Quadratic Time-Frequency Distributions (TFDs), on the other hand, allow the interpretation of representations from a much richer energy point of view (Hlawatsch and Boudreaux-Bartels, 1992; Loughlin et al., 1993).  Marginal properties express such interpretation. However, in view of the Uncertainty Principle, such properties are not sufficient to identify an energy density at each point on the time-frequency plane. (Ceravolo, 2009). A spectrogram, or quadratic representation of the amplitude of the STFT, is a three dimensional representation of the time-averaged Fourier Transform over adjacent time segments of a random process (Bendat and Piersol, 2011). It does not fulfil marginal properties and violates the Linearity Principle, generating crossed or interference terms. These spurious terms are restricted to those regions of the time-frequency domain where the auto-terms overlap. In the specific case of the spectrogram, for two sufficiently separated components in the time-frequency plane, its interference terms are practically nil (Ceravolo, 2009). Spectrograms can be used to obtain modal coordinates from the signals recorded from a mobile and a stationary sensor. These signals are divided into quasi-stationary segments and then the auto-spectral density and cross-spectral density functions for segments of both records are calculated. Finally  a modal identification procedure  is performed (Marulanda et al., 2016; Marulanda, J, 2014).

In this paper, the non-stationary signals analysis recorded from sensors is performed by using Cohen’s class TFDs. The adoption of a quadratic representation helps in overcoming the limitations from  time-frequency resolution, as these transformations, the energetic and the correlative, are not based on the segmentation of signals. (Cohen, 1995). Invariance to time and frequency shifts characterize the Cohen’s Class Distributions (Hlawatsch and Auger, 2013). The system response is recorded in the time-frequency plane as the evolution of the spectral components corresponding to the energy of individual vibration modes whenever the Cohen’s Class Distributions is in use (Bonato et al., 1998, 1999; Bonato et al., 2000; Bonato P. et al., 1997; Cohen, 1995; Hammond and White, 1996; Hlawatsch and Boudreaux-Bartels, 1992). The proposed technique assumes that the frequency range of the system input spans the vibration modes. As the input spectrum increases, so does the identification accuracy. The instantaneous amplitude ratios directly determine the modal amplitude ratios for the time-frequency representations of signals recorded from a mobile sensor and a stationary sensor. Additionally, the phase of the cross-time- frequency representation between the two signals estimate the phase relationships. The estimators retain their dependence on time since they are derived directly from the two-dimensional functions of frequency and time variables. modal responses in linear time invariant systems show a constant relationship between the amplitude and the phase, and, therefore, the identified mode shapes are characterized by their stability over time (Bonato et al., 1998).

==2. Theoretical Background==

''' Cohen’s Class Distributions'''

Cohen’s class distributions display a clear invariance of their representations for time and frequency shifts, a desirable property when correlating signal characteristics to phenomena occurring in the physical system generating the signal.  The following equation  describes all the Cohen’s class distributions (Hlawatsch and Auger, 2013):

{| class="formulaSCP" style="width: 100%;border-collapse: collapse;width: 100%;text-align: center;" 
|-
| 
{| style="text-align: center; margin:auto;width: 100%;"
|-
| <math>{D}_{ss}\left( t,f\right) =\int_{-\infty }^{+\infty }\int_{-\infty }^{+\infty }\int_{-\infty }^{+\infty }s\left( u+\frac{1}{2}\tau \right) {s}^{\ast }\left( u-\frac{1}{2}\tau \right) \Phi (\theta ,\tau ){e}^{-j2\pi \theta t-j2\pi \tau f+j2\pi \theta u}dud\tau d\theta</math> 
|}
|  style="text-align: right;width: 5px;text-align: right;white-space: nowrap;"|<span style="text-align: center; font-size: 75%;">(1)</span>
|}


<span style="text-align: center; font-size: 75%;">where </span> <math display="inline">s\left( t\right)</math> <span style="text-align: center; font-size: 75%;"> is the signal, </span> <math display="inline">{s}^{\ast }\left( t\right)</math> <span style="text-align: center; font-size: 75%;">is its complex conjugate, </span> <math display="inline">t</math><span style="text-align: center; font-size: 75%;"> is time, </span> <math display="inline">f</math><span style="text-align: center; font-size: 75%;"> is frequency and </span> <math display="inline">\tau</math> <span style="text-align: center; font-size: 75%;"> is the time delay, </span> <math display="inline">\omega =</math><math>2\pi f</math><span style="text-align: center; font-size: 75%;"> is the circular frequency; </span> <math display="inline">\Phi (\theta ,\tau )</math><span style="text-align: center; font-size: 75%;"> is the kernel of the distribution and </span> <math display="inline">D\left( t,f\right)</math> <span style="text-align: center; font-size: 75%;">is the Cohen Class TFD. The cross-time-frequency distribution between two signals </span> <math display="inline">s\left( t\right)</math> <span style="text-align: center; font-size: 75%;">and </span> <math display="inline">y\left( t\right)</math> <span style="text-align: center; font-size: 75%;">can be defined as follows:</span>

{| class="formulaSCP" style="width: 100%;border-collapse: collapse;width: 100%;text-align: center;" 
|-
| 
{| style="text-align: center; margin:auto;width: 100%;"
|-
| <math>{D}_{sy}\left( t,f\right) =\int_{-\infty }^{+\infty }\int_{-\infty }^{+\infty }\int_{-\infty }^{+\infty }s\left( u+\frac{1}{2}\tau \right) {y}^{\ast }\left( u-\frac{1}{2}\tau \right) \Phi (\theta ,\tau ){e}^{-j2\pi \theta t-j2\pi \tau f+j2\pi \theta u}dud\tau d\theta</math>


|}
|  style="text-align: right;width: 5px;text-align: right;white-space: nowrap;"|<span style="text-align: center; font-size: 75%;">(2)</span>
|}


<span style="text-align: center; font-size: 75%;">The Wigner–Ville distribution (WVD) is the basic TFD and is a quadratic form that measures a local time-frequency energy. This distribution is obtained from equation (1) using </span> <math display="inline">{\Phi }_{WV}\left( \theta ,\tau \right) =</math><math>1</math><span style="text-align: center; font-size: 75%;">. Although the WVD satisfies a several important mathematical properties (Bonato et al., 2001; Flandrin, 1984; Wang and McFadden, 1993), it is not very suitable for applications with multiple components signals since the bi-linearity of the transform causes the appearance of interference terms . These are spurious terms when using the time-frequency characteristics of the signal for modal identification. The smoothed pseudo-Wigner-Ville transform (SPWVD) results from </span> <math display="inline">{\Phi }_{SPWV}\left( \theta ,\tau \right) =</math><math>\pi {e}^{-{T}^{2}{\pi }^{2}{\theta }^{2}}{e}^{-{B}^{2}{\pi }^{2}{t}^{2}}</math><span style="text-align: center; font-size: 75%;">, where  </span> <math display="inline">B</math><span style="text-align: center; font-size: 75%;"> and  </span> <math display="inline">T</math><span style="text-align: center; font-size: 75%;"> are the standard deviations in time and frequency, respectively. With the exception of real value, time-shift-invariance, and frequency invariance, the application of the smoothed function causes the loss of most of the mathematical properties of the WVD, (Hlawatsch and Boudreaux-Bartels, 1992). The foregoing independence between windows makes the SPWVD very versatile for reducing cross—terms in the WVD. This property is useful for modal identification.The Choi-Williams Distribution (CWD) (Choi and Williams, 1989; Papandreou and Boudreaux-Bertels, 1993) is defined by </span> <math display="inline">{\Phi }_{CW}\left( \theta ,\tau \right) =</math><math>{e}^{\frac{-{\theta }^{2}{\tau }^{2}}{\sigma }}</math><span style="text-align: center; font-size: 75%;">. This kernel is defined in the ambiguity plane </span> <math display="inline">\left( \theta ,\tau \right)</math> <span style="text-align: center; font-size: 75%;">.  If </span> <math display="inline">\sigma</math> <span style="text-align: center; font-size: 75%;">is very large, the kernel vanishes, therefore leading to the WVD, otherwise, the kernel application will have a filtering effect. For small values of </span> <math display="inline">\sigma</math> <span style="text-align: center; font-size: 75%;">, the effect of the multiplication is to preserve the ambiguity function close the origin of the plane </span> <math display="inline">\left( \theta ,\tau \right)</math> <span style="text-align: center; font-size: 75%;">, therefore, this kernel satisfies the minimization property (Cohen, 1995). In multicomponent signals the authentic terms are usually close to the ambiguity plane axis, and the interference terms are dispersed away. The resulting characteristic function reduces the interference terms without significantly affecting the signal components (Bonato et al., 1997). Therefore, although the CWD violates the time and frequency support properties, it can be shown that the introduced error is generally negligible (Lippmann, 1989; Papandreou and Boudreaux-Bertels, 1993), so the </span> <math display="inline">\sigma</math> <span style="text-align: center; font-size: 75%;"> value should be adapted to each individual case, since the location of the interference terms and the signal components change with respect to the characteristics of the signal (Bonato et al., 1997). In this paper, SPWVD distribution is used. In this case, by parameterizing the windows, as they attenuate the interference terms without significantly modifying the signals, good modal identification results are obtained.</span>

''' Modal Identification Using Mobile Sensors'''

The proposed methodology requires a mobile sensor that acquires data continuously along the structure to be identified and simultaneous acquisition by a stationary sensor. Under ambient excitation using a zero mean Gaussian white noise process, the outputs of the system are also zero mean Gaussian processes, exciting the fundamental frequencies (Bendat and Piersol, 2011).  Since the recorded signal from the mobile sensor is non-stationary, a procedure based on the Cohen Class Distributions (i.e. SPWVD) is proposed. These distributions are invariable to time and frequency shifts, which is required for proper system identification.

<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">
<span style="text-align: center; font-size: 75%;"> [[Image:Marmolejo_et_al_2019b-image4.png|600px]] </span></div>

<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">
<span style="text-align: center; font-size: 75%;">Figure 3. Schematic description of the proposed setup on a simply supported beam.</span></div>

<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">
<span style="text-align: center; font-size: 75%;"> [[Image:Marmolejo_et_al_2019b-image5.png|600px]] </span></div>

<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">
<span style="text-align: center; font-size: 75%;">Figure 4. Equivalent model of the proposed setup showed in Fig. 3.</span></div>

<span style="text-align: center; font-size: 75%;">Assuming that the structure to be identified is the one-dimensional beam shown in Figure 3 has </span> <math display="inline">n</math><span style="text-align: center; font-size: 75%;">discrete degrees of freedom (''d.o.f.''), wich each ''d.o.f.'' corresponds to a segment of the signal from the mobile sensor (Figure 4). </span>

<span style="text-align: center; font-size: 75%;">Let </span> <math display="inline">{s}_{i}\left( t\right)</math> <span style="text-align: center; font-size: 75%;"> be the response (displacement) in the </span> <math display="inline">i</math><span style="text-align: center; font-size: 75%;">th position of the system, </span> <math display="inline">{d}^{\left( k\right) }(t)</math><span style="text-align: center; font-size: 75%;"> the response (displacement) associated with the </span> <math display="inline">k</math><span style="text-align: center; font-size: 75%;">th vibration mode, and  </span> <math display="inline">{\varnothing }_{i}^{\left( k\right) }</math><span style="text-align: center; font-size: 75%;"> a term of the normalized eigenvectors matrix, then </span> <math display="inline">{s}_{i}\left( t\right)</math> <span style="text-align: center; font-size: 75%;">can be in decouple form as,</span>

{| class="formulaSCP" style="width: 100%;border-collapse: collapse;width: 100%;text-align: center;" 
|-
| 
{| style="text-align: center; margin:auto;width: 100%;"
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| <math>{s}_{i}=\sum _{k}^{}{\varnothing }_{i}^{(k)}{d}^{(k)}~(t)</math>
|}
|  style="width: 5px;text-align: right;white-space: nowrap;"|<span style="text-align: center; font-size: 75%;">  (3)</span>
|}


<span style="text-align: center; font-size: 75%;">The term of the sum in equation (3) extends to the vibration modes that are detected in the </span> <math display="inline">i</math><span style="text-align: center; font-size: 75%;">th position corresponding to a discretized ''d.o.f.''   The structure under ambient excitation is equipped with a mobile sensor with constant velocity that continuously records the acceleration and, simultaneously, with a stationary sensor also recording the acceleration vibrations. For each </span> <math display="inline">ith</math><span style="text-align: center; font-size: 75%;">''d.o.f., ''the individual modal components  appear in the form of an energy peak in the time-frequency plane that can be written in complex form as (Bonato et al., 2000):</span>

{| class="formulaSCP" style="width: 100%;border-collapse: collapse;width: 100%;text-align: center;" 
|-
| 
{| style="text-align: center; margin:auto;width: 100%;"
|-
| <math>{s}_{i}^{(k)}={\varnothing }_{i}^{(k)}{d}^{(k)}~\left( t\right) ={\varnothing }_{i}^{\left( k\right) }{\tilde{d}}^{\left( k\right) }(t){e}^{j2\pi {f}^{(k)}t}</math>
|}
|  style="width: 5px;text-align: right;white-space: nowrap;"|<span style="text-align: center; font-size: 75%;">(4)</span>
|}


<span style="text-align: center; font-size: 75%;">where </span> <math display="inline">{\tilde{d}}^{\left( k\right) }(t)=</math><math>{A}^{\left( k\right) }(t){e}^{j{\varphi }^{\left( k\right) }(t)}</math><span style="text-align: center; font-size: 75%;"> is a baseband signal related to the </span> <math display="inline">kth</math><span style="text-align: center; font-size: 75%;"> mode;  </span> <math display="inline">{A}^{\left( k\right) }\left( t\right)</math> <span style="text-align: center; font-size: 75%;"> and </span> <math display="inline">{\varphi }^{\left( k\right) }(t)</math><span style="text-align: center; font-size: 75%;"> are the amplitude and phase modulation waveforms respectively; </span> <math display="inline">{f}^{(k)}</math><span style="text-align: center; font-size: 75%;"> is the natural frequency; </span> <math display="inline">{\varnothing }_{i}^{\left( k\right) }</math><span style="text-align: center; font-size: 75%;"> is the amplitude of the </span> <math display="inline">kth</math><span style="text-align: center; font-size: 75%;">  mode shape in the </span> <math display="inline">ith</math><span style="text-align: center; font-size: 75%;"> position. The complex form of equation (4) represents an analytical signal and, therefore, the spectrum of </span> <math display="inline">{d}^{(k)}~\left( t\right)</math> <span style="text-align: center; font-size: 75%;"> is single-sided. The WVD will transform the signal into:</span>

{| class="formulaSCP" style="width: 100%;border-collapse: collapse;width: 100%;text-align: center;" 
|-
| 
{| style="text-align: center; margin:auto;width: 100%;"
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| <math display="inline">{WVD}_{{s}_{i}^{\left( k\right) }{s}_{i}^{\left( k\right) }}\left( t,f\right) =</math><math>{\varnothing }_{i}^{{\left( k\right) }^{2}}{WVD}_{{\tilde{d}}^{\left( k\right) }{\tilde{d}}^{\left( k\right) }}\left( t,f\right) {\ast }_{f}\delta (f-</math><math>{f}^{(k)}</math><span style="text-align: center; font-size: 75%;">'')''</span>
|}
|  style="width: 5px;text-align: right;white-space: nowrap;"|<span style="text-align: center; font-size: 75%;">  (5)</span>
|}


<span style="text-align: center; font-size: 75%;">where </span> <math display="inline">{WVD}_{{\tilde{d}}^{\left( k\right) }{\tilde{d}}^{\left( k\right) }}</math><span style="text-align: center; font-size: 75%;"> is the WVD of the signal </span> <math display="inline">{\tilde{d}}^{\left( k\right) }(t)</math><span style="text-align: center; font-size: 75%;">, </span> <math display="inline">{\ast }_{f}</math><span style="text-align: center; font-size: 75%;"> is the convolution in frequency, and </span> <math display="inline">\delta (f-</math><math>{f}^{(k)}</math><span style="text-align: center; font-size: 75%;">'')'' is the Dirac delta (impulse) located in </span> <math display="inline">{f}^{(k)}</math><span style="text-align: center; font-size: 75%;">. In multi-component signals, when auto-terms and interference overlap, using a kernel capable of attenuating interference terms leads to an increased reliability of the estimated amplitude ratio. The use of highly selective kernels, ambiguity domain functions (Bonato et al., 2000), improves the capacity of the cross-time-frequency-transform to separate signal components, and therefore, allow amplitude ratio and phase difference estimators become close to each modal shape frequency. </span>

The following expression represents the Cohen’s Class Transforms:

{| class="formulaSCP" style="width: 100%;border-collapse: collapse;width: 100%;text-align: center;" 
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{| style="text-align: center; margin:auto;width: 100%;"
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| <math>{D}_{ss}\left( t,f\right) ={\varnothing }_{i}^{{\left( k\right) }^{2}}\Gamma \left( t,f\right) {{\ast }_{t,f}WVD}_{ss}\left( t,f\right)</math> 
|}
|  style="width: 5px;text-align: right;white-space: nowrap;"|<span style="text-align: center; font-size: 75%;">  (6)</span>
|}


<span style="text-align: center; font-size: 75%;">where </span> <math display="inline">{D}_{ss}\left( t,f\right)</math> <span style="text-align: center; font-size: 75%;"> is a Cohen class TFD, </span> <math display="inline">\Gamma \left( t,f\right)</math> <span style="text-align: center; font-size: 75%;"> is the kernel of the TFD in time-frequency plane.  </span> <math display="inline">{\ast }_{t,f}</math><span style="text-align: center; font-size: 75%;"> is the double convolution in time and frequency. From equation (6) it is evident that the energy of the transform concentrates close to the modal frequency </span> <math display="inline">{f}^{(k)}</math><span style="text-align: center; font-size: 75%;">. The distribution shape is determined by the time-frequency transform of the modulating waveform </span> <math display="inline">{\tilde{d}}^{\left( k\right) }(t)</math><span style="text-align: center; font-size: 75%;">. Because the shape of the modulated waveform is kept in the time–frequency domain. The amplitude ratio between two modal components can be determined from auto-time-frequency representations, as follows (Bonato et al., 1998; Bonato et al., 2000):</span>

{| class="formulaSCP" style="width: 100%;border-collapse: collapse;width: 100%;text-align: center;" 
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| 
{| style="text-align: center; margin:auto;width: 100%;"
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| <math>{\left. AR\left( t\right) \right| }_{f={f}^{(k)}}={\left. \sqrt{\frac{{D}_{{s}_{i}^{\left( k\right) }{s}_{i}^{\left( k\right) }}\left( t,f\right) }{{D}_{{s}_{j}^{\left( k\right) }{s}_{j}^{\left( k\right) }}\left( t,f\right) }}\right| }_{f={f}^{(k)}}=</math><math>\frac{{\varnothing }_{i}^{(k)}}{{\varnothing }_{j}^{(k)}}</math>
|}
|  style="width: 5px;text-align: right;white-space: nowrap;"|<span style="text-align: center; font-size: 75%;">(7)</span>
|}


<span style="text-align: center; font-size: 75%;">And the phase can be  deteremined from cross-time-frequency representtions, as follows:</span>

{| class="formulaSCP" style="width: 100%;border-collapse: collapse;width: 100%;text-align: center;" 
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{| style="text-align: center; margin:auto;width: 100%;"
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| <math>{\left. PH\left( t\right) \right| }_{f={f}^{\left( k\right) }}=phase{\left. \left\{ {D}_{{s}_{i}^{\left( k\right) }{s}_{j}^{\left( k\right) }}\left( t,f\right) \right\} \right| }_{f={f}^{\left( k\right) }}=</math><math>\Delta {\varphi }_{ij}^{(k)}</math>
|}
|  style="width: 5px;text-align: right;white-space: nowrap;"|<span style="text-align: center; font-size: 75%;">(8)</span>
|}


<span style="text-align: center; font-size: 75%;">The TFDs of the signals from both sensors, mobile and stationary, were used in separating modal and residual components and in estimating the modal Amplitudes Ratios (AR); while cross-time-frequency representations were used in estimating the Phase Ratios (PR) (Bonato et al., 1998). The system is displayed in the time-frequency plane as the unfolding of spectral components corresponding to the energy of the predominant vibration modes. The ARs, gathered from their time-frequency representations, are determined from the ratio between the instantaneous amplitudes of the signal segments. The PRs are the ratio between the imaginary and the real part of the cross- time-frequency representation of two signal segments (Bonato et al., 2000). Once the cross terms have been filtered, it is possible to identify a potential error source in the vibration modes due to a close coupling between the components. In contrast to classical frequency analysis, the cross time-frequency transform between channels can filter the products among various components. Although being closely coupled in frequency, they are uncorrelated in time (Bonato P. et al., 1997). </span>

<span style="text-align: center; font-size: 75%;">In the frequency ranges in which a single modal component is predominant, the estimators tend to be a constant value over time.  Because the variance (</span> <math display="inline">{\sigma }_{TFD}</math><span style="text-align: center; font-size: 75%;">) is a measure of the density of the time-frequency representation, the peaks of the variance of the cross-time-frequency transform are concentrated close to fundamental frequencies and, therefore, such frequencies are located in the maximum of the following function:</span>

{| class="formulaSCP" style="width: 100%;border-collapse: collapse;width: 100%;text-align: center;" 
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| 
{| style="text-align: center; margin:auto;width: 100%;"
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| <math>{\sigma }_{TFD}=\int_{0}^{{T}_{S}}{\left[ {D}_{{s}_{i}^{\left( k\right) }{s}_{j}^{\left( k\right) }}\left( t,f\right) -\overline{{D}_{{s}_{i}^{\left( k\right) }{s}_{j}^{\left( k\right) }}\left( t,f\right) }\right] }^{2}dt</math>
|}
|  style="text-align: right;width: 5px;text-align: right;white-space: nowrap;"|<span style="text-align: center; font-size: 75%;">(9)</span>
|}


<span style="text-align: center; font-size: 75%;">where  </span> <math display="inline">{T}_{S}</math><span style="text-align: center; font-size: 75%;"> is the length of the signal, and  </span> <math display="inline">\overline{{D}_{{s}_{i}^{\left( k\right) }{s}_{j}^{\left( k\right) }}\left( t,f\right) }</math><span style="text-align: center; font-size: 75%;">  indicates the mean value of the cross-time-frequency distribution.</span>

==3. Numerical Simulations==

<span style="text-align: center; font-size: 75%;">A proposed example to evaluate the described procedure using a numerical model of a simply supported beam (Figure 3). The beam’s </span> <math display="inline">n</math><span style="text-align: center; font-size: 75%;">th natural vibration mode shape and corresponding natural frequency are given by the following equation:</span>

{| class="formulaSCP" style="width: 100%;border-collapse: collapse;width: 100%;text-align: center;" 
|-
| 
{| style="text-align: center; margin:auto;width: 100%;"
|-
| <math display="inline">{\phi }_{n}\left( x\right) =\mathrm{sin}\,(\frac{n\pi x}{L})</math><span style="text-align: center; font-size: 75%;"> ; </span> <math display="inline">\, \, {\omega }_{n}=</math><math>\frac{{n}^{2}{\pi }^{2}}{{L}^{2}}\sqrt{\frac{EI}{{m}_{l}}}</math>
|}
|  style="width: 5px;text-align: right;white-space: nowrap;"|<span style="text-align: center; font-size: 75%;">(10)</span>
|}


<span style="text-align: center; font-size: 75%;">where </span> <math display="inline">L</math><span style="text-align: center; font-size: 75%;"> is the total length of the beam, </span> <math display="inline">E</math><span style="text-align: center; font-size: 75%;"> is the elasticity modulus of the material, </span> <math display="inline">I</math><span style="text-align: center; font-size: 75%;"> is the moment of inertia of the cross section, and </span> <math display="inline">{m}_{l}</math><span style="text-align: center; font-size: 75%;"> is the constant mass per unit length (Chopra, 2012). In the following numerical simulations. In this simulation is assumed that </span> <math display="inline">L=</math><math>60\, m</math><span style="text-align: center; font-size: 75%;">, </span> <math display="inline">E=</math><math>25\, GPa</math><span style="text-align: center; font-size: 75%;">, </span> <math display="inline">I=</math><math>6.75\, {m}^{4}</math><span style="text-align: center; font-size: 75%;">, </span> <math display="inline">{m}_{l}=</math><math>150\, \frac{kN}{\frac{g}{m}}</math><span style="text-align: center; font-size: 75%;">, and </span> <math display="inline">{\zeta }_{n}=</math><math>0.05</math><span style="text-align: center; font-size: 75%;">; and, therefore, the first three natural frequencies of the beam are </span> <math display="inline">1.45</math><span style="text-align: center; font-size: 75%;">, </span> <math display="inline">5.80</math><span style="text-align: center; font-size: 75%;">, and </span> <math display="inline">13.05\, Hz</math><span style="text-align: center; font-size: 75%;">. The stationary sensor was located at </span> <math display="inline">25\, m</math><span style="text-align: center; font-size: 75%;">from the left support and the mobile sensor had a velocity of </span> <math display="inline">\, 0.0167\frac{m}{s}</math><span style="text-align: center; font-size: 75%;">. The excitation of the beam corresponds to a Gaussian white noise acceleration at the supports. The simulated response signals were sampled at </span> <math display="inline">600\, Hz</math><span style="text-align: center; font-size: 75%;">, and then conditioned using a Hilbert transform and sub-sampled at </span> <math display="inline">30\, Hz</math><span style="text-align: center; font-size: 75%;">. SPWVD was used for the modal identification. The computed SPWVD was performed with a Hanning time smoothing window of </span> <math display="inline">39</math><span style="text-align: center; font-size: 75%;"> samples, Hanning frequency smoothing window of </span> <math display="inline">125</math><span style="text-align: center; font-size: 75%;"> samples and 256 frequency bins. The identified mode shapes </span> <math display="inline">{\Phi }_{j}</math><span style="text-align: center; font-size: 75%;">are compared to the exact mode shapes </span> <math display="inline">{\Phi }_{k}</math><span style="text-align: center; font-size: 75%;">using the Modal Assurance Criterion (MAC) via  equation (11) (Pastor et al., 2012). The damping is identified using the Half-Power Bandwidth method (Bonato P. et al., 1997; Chopra, 2012). The influence of noise in the identification is evaluated. The software used for signals processing using TFDS was The Time-Frequency Toolbox (TFTB) (Auger F. et al., 1996). </span>

{| class="formulaSCP" style="width: 100%;margin: 1em auto 0.1em auto;border-collapse: collapse;width: 100%;text-align: center;" 
|-
| 
{| style="text-align: center; margin:auto;width: 100%;"
|-
| <math>{MAC}_{jk}=\frac{{({\Phi }_{j}^{T}{\Phi }_{k})}^{2}}{({\Phi }_{j}^{T}{\Phi }_{j})({\Phi }_{k}^{T}{\Phi }_{k})}</math>
|}
|  style="width: 5px;text-align: right;white-space: nowrap;"|<span style="text-align: center; font-size: 75%;">(11)</span>
|}


<span style="text-align: center; font-size: 75%;">'''TFD-based mode shapes identification'''</span>

<span id='_Hlk488654953'></span><span id='_Hlk488655038'>Figure 5 shows the time history of the signal recorded with the mobile sensor traveling at 0.0167 m/s, along with the spectral density and auto-SPWVD. Figure 6 shows the time history of the signal recorded with the stationary sensor, along with its spectral density and auto-SPWVD. Figure 7 shows the time history of the signals recorded with the mobile and the stationary sensors along with their spectral density and cross-SPWVD. Figure 8 shows the standard deviation of the cross-SPWVD. The modal frequencies and damping ratios were obtained via Equation (9) by selecting peaks and using the Half-Power Bandwidth method (Figure 8). The mode shapes were obtained from TFDs (i.e. SPWVD), as given by Equations (7) and (8). Table 1 summarizes results of the modal parameter identification, where it is shown that errors in natural frequencies are generally less than 1.5 %. The identified mode shapes are also compared to the exact mode shapes. In this example, 59 modal coordinates were identified, and MAC values were all equal to one for the first three modes (Figure 9). MAC values for different mobile sensor velocity values, ranging from </span> <math display="inline">0.0167\frac{m}{s}</math><span style="text-align: center; font-size: 75%;"> to </span> <math display="inline">0.25\frac{m}{s}</math><span style="text-align: center; font-size: 75%;">, were calculated. Figure 10 shows the variations in the identified modal coordinates (IMC) per unity of beam’s length  versus  the mobile sensor’s velocity, it is assumed that two modes are considered correlated when the MAC value is  equal to 0.95, which corresponds to an angle of  18 degrees (Pastor et al., 2012). From Figure 11, it can be observed that good identification results have been obtained from the simulated ambient response data contaminated with different intensity levels of white gaussian noise.</span>

<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">
<span style="text-align: center; font-size: 75%;">Table 1. Summary of results obtained using equations (7), (8), (9) and (11)</span></div>

{| style="width: 100%;margin: 1em auto 0.1em auto;border-collapse: collapse;" 
|-
|  rowspan='2' style="border-top: 1pt solid black;text-align: center;vertical-align: top;"|<span style="text-align: center; font-size: 75%;">MODE</span>
|  rowspan='2' style="border-top: 1pt solid black;text-align: center;vertical-align: top;"|<span style="text-align: center; font-size: 75%;">Theoretical frequencies</span>

<span style="text-align: center; font-size: 75%;">''(Hz)''</span>
|  colspan='3'  style="border-top: 1pt solid black;text-align: center;vertical-align: top;"|<span style="text-align: center; font-size: 75%;">Identification by TFD (SPWVD)</span>
|  rowspan='2' style="border-top: 1pt solid black;text-align: center;vertical-align: top;"|<span style="text-align: center; font-size: 75%;">Frequency Error</span>

<span style="text-align: center; font-size: 75%;">(%)</span>
|-
|  style="text-align: center;vertical-align: top;"|<span style="text-align: center; font-size: 75%;">'''Natural '''</span>

<span style="text-align: center; font-size: 75%;">'''frequencies (Hz)'''</span>
|  style="text-align: center;vertical-align: top;"|<span style="text-align: center; font-size: 75%;">'''Damping'''</span>

<span style="text-align: center; font-size: 75%;">'''ratios'''</span>
|  style="text-align: center;vertical-align: top;"|<span style="text-align: center; font-size: 75%;">'''MAC'''</span>

<span style="text-align: center; font-size: 75%;">'''values'''</span>
|-
|  style="text-align: center;vertical-align: top;"|<span style="text-align: center; font-size: 75%;">1</span>
|  style="text-align: center;vertical-align: top;"|<span style="text-align: center; font-size: 75%;">1.450</span>
|  style="text-align: center;vertical-align: top;"|<span style="text-align: center; font-size: 75%;">1.465</span>
|  style="text-align: center;vertical-align: top;"|<span style="text-align: center; font-size: 75%;">0.060</span>
|  style="text-align: center;vertical-align: top;"|<span style="text-align: center; font-size: 75%;">1.00</span>
|  style="text-align: center;vertical-align: top;"|<span style="text-align: center; font-size: 75%;">1.03</span>
|-
|  style="text-align: center;vertical-align: top;"|<span style="text-align: center; font-size: 75%;">2</span>
|  style="text-align: center;vertical-align: top;"|<span style="text-align: center; font-size: 75%;">5.798</span>
|  style="text-align: center;vertical-align: top;"|<span style="text-align: center; font-size: 75%;">5.801</span>
|  style="text-align: center;vertical-align: top;"|<span style="text-align: center; font-size: 75%;">0.029</span>
|  style="text-align: center;vertical-align: top;"|<span style="text-align: center; font-size: 75%;">1.00</span>
|  style="text-align: center;vertical-align: top;"|<span style="text-align: center; font-size: 75%;">0.05</span>
|-
|  style="border-bottom: 1pt solid black;text-align: center;vertical-align: top;"|<span style="text-align: center; font-size: 75%;">3</span>
|  style="border-bottom: 1pt solid black;text-align: center;vertical-align: top;"|<span style="text-align: center; font-size: 75%;">13.046</span>
|  style="border-bottom: 1pt solid black;text-align: center;vertical-align: top;"|<span style="text-align: center; font-size: 75%;">13.008</span>
|  style="border-bottom: 1pt solid black;text-align: center;vertical-align: top;"|<span style="text-align: center; font-size: 75%;">0.025</span>
|  style="border-bottom: 1pt solid black;text-align: center;vertical-align: top;"|<span style="text-align: center; font-size: 75%;">1.00</span>
|  style="border-bottom: 1pt solid black;text-align: center;vertical-align: top;"|<span style="text-align: center; font-size: 75%;">0.29</span>
|}


<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">
<span style="text-align: center; font-size: 75%;"> [[Image:Marmolejo_et_al_2019b-image6.jpeg|378px]] </span></div>

<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">
<span style="text-align: center; font-size: 75%;">Figure 5. Signal recorded with the mobile sensor: time history, spectral density and SPWVD</span></div>

<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">
<span style="text-align: center; font-size: 75%;"> [[Image:Marmolejo_et_al_2019b-image7.jpeg|378px]] </span></div>

<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">
<span style="text-align: center; font-size: 75%;">Figure 6. Signal recorded with the stationary sensor: time history, spectral density and SPWVD</span></div>

<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">
<span style="text-align: center; font-size: 75%;"> [[Image:Marmolejo_et_al_2019b-image8.jpeg|378px]] </span></div>

<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">
<span style="text-align: center; font-size: 75%;">Figure 7. Signals recorded with the stationary and the mobile sensors: time history, spectral density and SPWV cross-distribution.</span></div>

<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">
<span style="text-align: center; font-size: 75%;"> [[Image:Marmolejo_et_al_2019b-image9.jpeg|378px]] </span></div>

<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">
<span style="text-align: center; font-size: 75%;">Figure 8. Standard deviation from the SPWVD cross-distribution.</span></div>

<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">
<span style="text-align: center; font-size: 75%;"> [[Image:Marmolejo_et_al_2019b-image10.jpeg|378px]] </span></div>

<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">
<span style="text-align: center; font-size: 75%;">Figure 9.  First three mode shapes identified using equations (7) and (8): 59 modal coordinates were identified.  </span></div>

<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">
<span style="text-align: center; font-size: 75%;"> [[Image:Marmolejo_et_al_2019b-image11.jpeg|378px]] </span></div>

<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">
<span style="text-align: center; font-size: 75%;">Figure 10. IMC/m values versus mobile sensor velocity.</span></div>

'''Signal-to-Noise Ratio'''

In most experimental applications, the acceleration record from the mobile sensor would be more contaminated with noise than the stationary sensor due to the motion system. To evaluate the effect of such noise on the identified modes through the MAC values, artificial normally distributed white noise was added to the acceleration response of the mobile to generate signal-to-noise ratios from -5 to 20 dB. Figure 11 shows the MAC values versus SNR for the first three identified modes. It can be seen from Figure 11 that the identification accuracy of all mode shapes can be maintained at a high level using the proposed method for SNR values higher than 15 dB.

<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">
 [[Image:Marmolejo_et_al_2019b-image12.jpeg|378px]] </div>

<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">
<span style="text-align: center; font-size: 75%;">Figure 11. MAC values of the first three identified modes for variation in the signal-to-noise ratio.</span></div>

==4. CONCLUSIONS==

This paper presents the use of mobile sensors for modal identification through Cohen’s Class TFDs, providing identified modal shapes with high spatial resolution. Mobile sensors can be used in structures and mechanical systems under normal operating conditions and unknown excitation. It is envisaged that mobile sensors not only offer flexible but adaptable spatial resolutions that may enhance the theory and practice of future SHM and damage detection research and technological development.

The mobile sensor continuously records vibration data as it travels along trajectories on the structure, therefore, modal coordinates are identified with a higher spatial density than using stationary sensors. The quantity of identified modal coordinates depends on the velocity of the sensor, the selected kernels, and the identified frequencies.

The TFD-based modal identification utilized to identify the structural system in terms of natural frequencies, damping ratios and mode shapes, along with the use of mobile sensors complied successfully. The mode shapes are estimated through the auto and cross time-frequency transforms of the signals from mobile and stationary sensors (i.e. modulating waveforms). The proposed methodology is highly accurate when calculating mode shapes by determining the amplitude and phase ratios between TFDs of signals recorded with the mobile and the stationary sensors, respectively. The standard deviations of cross time-frequency transforms of the recorded signals are related to modal frequencies and damping ratios.

<span style="text-align: center; font-size: 75%;">The effectiveness of the technique is clear, based on auto and cross-time-frequency estimators for structural identification when non-stationary signals are used. For velocities between </span> <math display="inline">0.25\frac{m}{s}</math><span style="text-align: center; font-size: 75%;"> to </span> <math display="inline">0.25\frac{m}{s}</math><span style="text-align: center; font-size: 75%;">,  the variations in the identified modal coordinates per unity of beam’s length ranges from </span> <math display="inline">20\, IMC/m</math><span style="text-align: center; font-size: 75%;"> to </span> <math display="inline">1.5\, IMC/m</math><span style="text-align: center; font-size: 75%;">. For a velocity of </span> <math display="inline">0.0167\frac{m}{s}</math><span style="text-align: center; font-size: 75%;">, 59 modal coordinates were identified, and MAC values were all equal to one for the first three modes using only two acceleration records. For signal-to-noise ratios higher than 15 dB, all identified modal shapes are noise stable. </span>

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